I can give you a tl;dr

1. You and the person you want to zap both need a lightning address in your profiles.

2. Query their lightning address over HTTP to find their "callback" URL.

3. Sign a Zap Request event, and POST it to their "callback" URL.

That's it.

If you have enough funds, the Lightning server will transfer the money to the recipient and publish a Zap Receipt event onto relays, confirming the transaction.

The Lightning server does most of the heavy lifting.